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Theorem onsucuni2 6675
Description: A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
onsucuni2  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  A )

Proof of Theorem onsucuni2
StepHypRef Expression
1 eleq1 2495 . . . . . 6  |-  ( A  =  suc  B  -> 
( A  e.  On  <->  suc 
B  e.  On ) )
21biimpac 488 . . . . 5  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  B  e.  On )
3 eloni 5452 . . . . 5  |-  ( suc 
B  e.  On  ->  Ord 
suc  B )
4 ordsuc 6655 . . . . . . . 8  |-  ( Ord 
B  <->  Ord  suc  B )
5 ordunisuc 6673 . . . . . . . 8  |-  ( Ord 
B  ->  U. suc  B  =  B )
64, 5sylbir 216 . . . . . . 7  |-  ( Ord 
suc  B  ->  U. suc  B  =  B )
7 suceq 5507 . . . . . . 7  |-  ( U. suc  B  =  B  ->  suc  U. suc  B  =  suc  B )
86, 7syl 17 . . . . . 6  |-  ( Ord 
suc  B  ->  suc  U. suc  B  =  suc  B
)
9 ordunisuc 6673 . . . . . 6  |-  ( Ord 
suc  B  ->  U. suc  suc 
B  =  suc  B
)
108, 9eqtr4d 2466 . . . . 5  |-  ( Ord 
suc  B  ->  suc  U. suc  B  =  U. suc  suc 
B )
112, 3, 103syl 18 . . . 4  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. suc  B  =  U. suc  suc  B
)
12 unieq 4227 . . . . . 6  |-  ( A  =  suc  B  ->  U. A  =  U. suc  B )
13 suceq 5507 . . . . . 6  |-  ( U. A  =  U. suc  B  ->  suc  U. A  =  suc  U. suc  B
)
1412, 13syl 17 . . . . 5  |-  ( A  =  suc  B  ->  suc  U. A  =  suc  U.
suc  B )
15 suceq 5507 . . . . . 6  |-  ( A  =  suc  B  ->  suc  A  =  suc  suc  B )
1615unieqd 4229 . . . . 5  |-  ( A  =  suc  B  ->  U. suc  A  =  U. suc  suc  B )
1714, 16eqeq12d 2444 . . . 4  |-  ( A  =  suc  B  -> 
( suc  U. A  = 
U. suc  A  <->  suc  U. suc  B  =  U. suc  suc  B ) )
1811, 17syl5ibr 224 . . 3  |-  ( A  =  suc  B  -> 
( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  U. suc  A ) )
1918anabsi7 826 . 2  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  = 
U. suc  A )
20 eloni 5452 . . . 4  |-  ( A  e.  On  ->  Ord  A )
21 ordunisuc 6673 . . . 4  |-  ( Ord 
A  ->  U. suc  A  =  A )
2220, 21syl 17 . . 3  |-  ( A  e.  On  ->  U. suc  A  =  A )
2322adantr 466 . 2  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  U. suc  A  =  A )
2419, 23eqtrd 2463 1  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872   U.cuni 4219   Ord word 5441   Oncon0 5442   suc csuc 5444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-tr 4519  df-eprel 4764  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-ord 5445  df-on 5446  df-suc 5448
This theorem is referenced by:  rankxplim3  8360  rankxpsuc  8361
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